1D Kinematics Basics

Displacement, Position, and Time

Kinematics is the study of change, position, and time. It focuses on describing how objects move without considering the forces causing the motion.

• Displacement: Change in position; a vector quantity indicating how far and in what direction an object has moved.

• Position: The location of an object at a given time.

• Time: The duration over which motion occurs.

• Velocity: The rate of change of displacement; a vector quantity.

• Acceleration: The rate at which velocity changes with time.

Example: If an object moves from position m to m, the displacement is m.

Distance vs. Displacement

Distance is the total length of the path traveled, regardless of direction, while displacement is the straight-line change in position.

• Distance: Scalar quantity; always positive.

• Displacement: Vector quantity; can be positive or negative.

Velocity and Speed

Definitions and Formulas

Velocity and speed are both measures of how fast an object moves, but velocity includes direction.

• Average velocity:

• Average speed:

Example: If an object moves 10 m north and 5 m east, the magnitude of displacement is found using the Pythagorean theorem: m.

Acceleration

Definition and Calculation

Acceleration is the rate at which velocity changes. It is a vector quantity and can be positive (speeding up) or negative (slowing down).

• Average acceleration:

• Where is the change in velocity and is the change in time.

Object Displacement and Velocity Equations

Key kinematic equations relate displacement, velocity, acceleration, and time:

Graphical Analysis of Motion

Displacement vs. Time and Velocity vs. Time Graphs

Graphs are essential tools for visualizing motion. The slope of a displacement vs. time graph gives velocity, while the slope of a velocity vs. time graph gives acceleration.

• Displacement vs. Time: Slope = velocity

• Velocity vs. Time: Slope = acceleration

Area Under Graphs

The area under a velocity vs. time graph represents displacement, and the area under an acceleration vs. time graph represents change in velocity.

• For constant acceleration, area is a rectangle or triangle.

• For variable acceleration, area can be calculated using integration or summing smaller areas.

Motion Scenarios

Common Graph Shapes

Different motion scenarios produce characteristic graph shapes:

• Object at rest: Flat line on velocity vs. time graph.

• Object moving at constant velocity: Horizontal line above zero.

• Object accelerating uniformly: Straight line with positive or negative slope.

Free Fall

Motion Under Gravity

Objects in free fall experience constant acceleration due to gravity ( m/s2 downward). The equations of motion for free fall are:

Examples and Graphs

Example: A ball thrown upward with initial velocity will reach a maximum height where , then fall back down.

• Time in air, maximum height, and velocity at different points can be calculated using kinematic equations.

Projectile Motion

Two-Dimensional Motion

Projectile motion involves both horizontal and vertical components. The horizontal motion has constant velocity, while the vertical motion has constant acceleration due to gravity.

• Horizontal displacement:

• Vertical displacement:

• Maximum height:

• Range: (for symmetric projectile motion)

Velocity Components and Example Problems

Velocity is resolved into x and y components using trigonometry:

Example: A ball thrown at an angle with initial velocity will have both horizontal and vertical motion, and its trajectory can be analyzed using the above equations.

Summary Table: Kinematic Equations